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URL: https://oeis.org/A239571

⇱ A239571 - OEIS

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A239571
Number of ways to place 5 points on a triangular grid of side n so that no two of them are adjacent.
8
0, 0, 27, 999, 11565, 74811, 342042, 1239525, 3799488, 10259640, 25076952, 56552364, 119324403, 238062357, 452774595, 826245798, 1454229216, 2479147536, 4108199481, 6636929805, 10479498849, 16207085223, 24596072424, 36687908235, 53862785520, 77929575480
OFFSET
3,3
COMMENTS
Rotations and reflections of placements are counted. If they are to be ignored see A239575.
a(n) is the coefficient of x^5 in the independence polynomial of the (n-1)-triangular grid graph. - Eric W. Weisstein, Mar 27 2026
LINKS
Eric Weisstein's World of Mathematics, Independence Polynomial.
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1)
FORMULA
a(n) = (n -3) * (n -4) * (n^8 +12*n^7 -58*n^6 -860*n^5 +2141*n^4 +23728*n^3 -61316*n^2 -244928*n +770880)/3840.
G.f.: -3*x^5*(40*x^8-185*x^7+198*x^6+213*x^5-243*x^4-638*x^3+687*x^2+234*x+9) / (x-1)^11. - Colin Barker, Mar 22 2014
a(n) = 11*a(n-1)-55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+330*a(n-7)-165*a(n-8)+55*a(n-9)-11*a(n-10)+a(n-11). - Eric W. Weisstein, Mar 27 2026
MATHEMATICA
CoefficientList[Series[-3 x^2 (40 x^8 - 185 x^7 + 198 x^6 + 213 x^5 - 243 x^4 - 638 x^3 + 687 x^2 + 234 x + 9)/(x - 1)^11, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 23 2014 *)
Table[(n - 4) (n - 3) (n^8 + 12 n^7 - 58 n^6 - 860 n^5 + 2141 n^4 + 23728 n^3 - 61316 n^2 - 244928 n + 770880)/3840, {n, 3, 20}] (* Eric W. Weisstein, Mar 27 2026 *)
LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {0, 0, 27, 999, 11565, 74811, 342042, 1239525, 3799488, 10259640, 25076952}, 20] (* Eric W. Weisstein, Mar 27 2026 *)
PROG
(PARI) concat([0, 0], Vec(-3*x^5*(40*x^8-185*x^7+198*x^6+213*x^5-243*x^4-638*x^3+687*x^2+234*x+9)/(x-1)^11 + O(x^100))) \\ Colin Barker, Mar 22 2014
(Magma) [(n^2-7*n+12)*(n^8+12*n^7-58*n^6-860*n^5+2141*n^4 +23728*n^3-61316*n^2-244928*n+770880)/3840: n in [3..40]]; // Vincenzo Librandi, Mar 23 2014
CROSSREFS
Cf. A239567, A239575, A239568 (2 points), A239569 (3 points), A239570 (4 points), A282998 (6 points).
Sequence in context: A129999 A132059 A292362 * A017019 A143366 A143705
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, Mar 22 2014
STATUS
approved